Informative Plots

Time Series

If we see a trend, the time series is not stationary (i.e. does not depend on the time of the observation).

Bar plot

Modified Series

\[ R_i = \frac{X_i - X_{i-7}}{X_{i-7}} * 100 \] \(R_i\) is the relative weekly change (increase or decrease) in percentage.

Peaks-Over-Threshold

  1. Examine the suitability for each time series \(R_i\) separately:

  2. Explain your choice of threshold and briefly comment on the diagnostic plots for each model:

Elizabeth II

Distribution

Mean Residual Plot

The selected threshold \(u\) is 125.

Peaks-Over-Threshold Plot

United States

Distribution

Mean Residual Plot

The selected threshold \(u\) is 35.

Peaks-Over-Threshold Plot

Queen Victoria

Distribution

Mean Residual Plot

The selected threshold \(u\) is 100.

Peaks-Over-Threshold Plot

World War II

Distribution

Mean Residual Plot

The selected threshold \(u\) is 30.

Peaks-Over-Threshold Plot

World War I

Distribution

Mean Residual Plot

The selected threshold \(u\) is 25.

Peaks-Over-Threshold Plot

George VI

Distribution

Mean Residual Plot

The selected threshold \(u\) is 200.

Peaks-Over-Threshold Plot

United Kingdom

Distribution

Mean Residual Plot

The selected threshold \(u\) is 50.

Peaks-Over-Threshold Plot

Princess Margaret, Countess of Snowdon

Distribution

Mean Residual Plot

The selected threshold \(u\) is 700.

Peaks-Over-Threshold Plot

Prince Philip, Duke of Edinburgh

Distribution

Mean Residual Plot

The selected threshold \(u\) is 100.

Peaks-Over-Threshold Plot

Winston Churchill

Distribution

Mean Residual Plot

The selected threshold \(u\) is 100.

Peaks-Over-Threshold Plot

Diana, Princess of Wales

Distribution

Mean Residual Plot

The selected threshold \(u\) is 70.

Peaks-Over-Threshold Plot

2016 Summer Olympics

Distribution

Mean Residual Plot

The selected threshold \(u\) is 60.

Peaks-Over-Threshold Plot

Suitability of POT: - Princess Margaret, United Kingdom, United States (?), Wiston Churchill do not seem to be suitable for POT because of the too few numbers of exceedances.

  • Estimate the 99%-quantile of each series and give a corresponding measure of uncertainty = provide an interval of values within which the true value of the threshol is believed to lie with a stated probability (MSE ? with confidence interval ?)
##             2016_Summer_Olympics Diana,_Princess_of_Wales Elizabeth_II
## quantile99              143.5829                 363.1843     642.9873
## uncertainty             179.0603                 492.1521    1753.4239
##              George_VI Prince_Philip,_Duke_of_Edinburgh
## quantile99    520.8474                         1111.279
## uncertainty 10635.0359                         1579.921
##             Princess_Margaret,_Countess_of_Snowdon Queen_Victoria
## quantile99                                973.8326       421.7058
## uncertainty                             17545.2920       544.1161
##             United_Kingdom United_States Winston_Churchill World_War_I
## quantile99        79.08561      62.24281          846.2276    94.64611
## uncertainty     1784.17935     266.51113         7139.2646    77.44259
##             World_War_II
## quantile99      61.75727
## uncertainty     48.12695

Detecting Simultaneous High Load

Graphical Method

Numerical Method

2016_Summer_Olympics Diana,_Princess_of_Wales Elizabeth_II George_VI Prince_Philip,_Duke_of_Edinburgh Princess_Margaret,_Countess_of_Snowdon Queen_Victoria United_Kingdom United_States Winston_Churchill World_War_I World_War_II
2016_Summer_Olympics 1.0000000 0.0714286 0.0000000 0.0714286 0.0357143 0.1071429 0.0357143 0.1071429 0.2142857 0.0357143 0.0000000 0.0357143
Diana,_Princess_of_Wales 0.0714286 1.0000000 0.0714286 0.0714286 0.1428571 0.0714286 0.0714286 0.0357143 0.0357143 0.0000000 0.0000000 0.0357143
Elizabeth_II 0.0000000 0.0714286 1.0000000 0.4642857 0.6071429 0.4642857 0.2857143 0.1071429 0.0000000 0.0357143 0.1428571 0.0000000
George_VI 0.0714286 0.0714286 0.4642857 1.0000000 0.5357143 0.5714286 0.3214286 0.0714286 0.0357143 0.0714286 0.1071429 0.0357143
Prince_Philip,_Duke_of_Edinburgh 0.0357143 0.1428571 0.6071429 0.5357143 1.0000000 0.5357143 0.2142857 0.0714286 0.0000000 0.0357143 0.1071429 0.0357143
Princess_Margaret,_Countess_of_Snowdon 0.1071429 0.0714286 0.4642857 0.5714286 0.5357143 1.0000000 0.2142857 0.0714286 0.0714286 0.0000000 0.1071429 0.0357143
Queen_Victoria 0.0357143 0.0714286 0.2857143 0.3214286 0.2142857 0.2142857 1.0000000 0.0357143 0.0000000 0.0000000 0.1071429 0.0000000
United_Kingdom 0.1071429 0.0357143 0.1071429 0.0714286 0.0714286 0.0714286 0.0357143 1.0000000 0.0357143 0.1785714 0.1071429 0.1071429
United_States 0.2142857 0.0357143 0.0000000 0.0357143 0.0000000 0.0714286 0.0000000 0.0357143 1.0000000 0.0714286 0.1785714 0.1785714
Winston_Churchill 0.0357143 0.0000000 0.0357143 0.0714286 0.0357143 0.0000000 0.0000000 0.1785714 0.0714286 1.0000000 0.1071429 0.0714286
World_War_I 0.0000000 0.0000000 0.1428571 0.1071429 0.1071429 0.1071429 0.1071429 0.1071429 0.1785714 0.1071429 1.0000000 0.4285714
World_War_II 0.0357143 0.0357143 0.0000000 0.0357143 0.0357143 0.0357143 0.0000000 0.1071429 0.1785714 0.0714286 0.4285714 1.0000000